Take with and its partners

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch Orbital angular momentum, the spherical harmonics and rotations

Problem

Take

\hat L_z = \hat x\hat p_y - \hat y\hat p_x

with

\hat p_x = -i\hbar\,\partial/\partial x

and its partners, acting on smooth functions of

x

y

z

. You may quote

[\hat x,\hat p_x] = i\hbar

, that position and momentum operators referring to different axes commute, that

\hat p_x\hat p_y = \hat p_y\hat p_x

for smooth functions, and the product rule

[\hat A,\hat B\hat C] = \hat B[\hat A,\hat C] + [\hat A,\hat B]\hat C

. (a) Show that

[\hat L_z,\hat p_x] = i\hbar\hat p_y

. (b) Test the identity on the particular function

f = y^2

, evaluating

\hat L_z

with its spherical form

-i\hbar\,\partial/\partial\phi

and

y = r\sin\theta\sin\phi

, and

\hat p_x

\hat p_y

in Cartesian form. Show that both sides give the same function.